| Mathbox for Zhi Wang |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > oppcthin | Structured version Visualization version GIF version | ||
| Description: The opposite category of a thin category is thin. (Contributed by Zhi Wang, 29-Sep-2024.) |
| Ref | Expression |
|---|---|
| oppcthin.o | ⊢ 𝑂 = (oppCat‘𝐶) |
| Ref | Expression |
|---|---|
| oppcthin | ⊢ (𝐶 ∈ ThinCat → 𝑂 ∈ ThinCat) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oppcthin.o | . . . 4 ⊢ 𝑂 = (oppCat‘𝐶) | |
| 2 | eqid 2765 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
| 3 | 1, 2 | oppcbas 17762 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝑂) |
| 4 | 3 | a1i 11 | . 2 ⊢ (𝐶 ∈ ThinCat → (Base‘𝐶) = (Base‘𝑂)) |
| 5 | eqidd 2766 | . 2 ⊢ (𝐶 ∈ ThinCat → (Hom ‘𝑂) = (Hom ‘𝑂)) | |
| 6 | simpl 487 | . . . 4 ⊢ ((𝐶 ∈ ThinCat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐶 ∈ ThinCat) | |
| 7 | simprr 784 | . . . 4 ⊢ ((𝐶 ∈ ThinCat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶)) | |
| 8 | simprl 782 | . . . 4 ⊢ ((𝐶 ∈ ThinCat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶)) | |
| 9 | eqid 2765 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
| 10 | 6, 7, 8, 2, 9 | thincmo 50058 | . . 3 ⊢ ((𝐶 ∈ ThinCat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ∃*𝑓 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) |
| 11 | 9, 1 | oppchom 17759 | . . . . 5 ⊢ (𝑥(Hom ‘𝑂)𝑦) = (𝑦(Hom ‘𝐶)𝑥) |
| 12 | 11 | eleq2i 2857 | . . . 4 ⊢ (𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ↔ 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) |
| 13 | 12 | mobii 2578 | . . 3 ⊢ (∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥)) |
| 14 | 10, 13 | sylibr 237 | . 2 ⊢ ((𝐶 ∈ ThinCat ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝑂)𝑦)) |
| 15 | thincc 50052 | . . 3 ⊢ (𝐶 ∈ ThinCat → 𝐶 ∈ Cat) | |
| 16 | 1 | oppccat 17766 | . . 3 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) |
| 17 | 15, 16 | syl 18 | . 2 ⊢ (𝐶 ∈ ThinCat → 𝑂 ∈ Cat) |
| 18 | 4, 5, 14, 17 | isthincd 50066 | 1 ⊢ (𝐶 ∈ ThinCat → 𝑂 ∈ ThinCat) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ∃*wmo 2567 ‘cfv 6525 (class class class)co 7400 Basecbs 17257 Hom chom 17309 Catccat 17708 oppCatcoppc 17755 ThinCatcthinc 50047 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5231 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5105 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6291 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-tpos 8210 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-er 8682 df-en 8932 df-dom 8933 df-sdom 8934 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12222 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12493 df-z 12580 df-dec 12700 df-sets 17212 df-slot 17230 df-ndx 17242 df-base 17258 df-hom 17322 df-cco 17323 df-cat 17712 df-cid 17713 df-oppc 17756 df-thinc 50048 |
| This theorem is referenced by: oduoppcciso 50196 |
| Copyright terms: Public domain | W3C validator |